Therefore, the magnitude of the four-acceleration (which is an invariant scalar) is equal to the proper acceleration that a moving particle "feels" moving along a world line. The world lines having constant magnitude of four-acceleration are Minkowski-circles i.e. hyperbolas (see hyperbolic motion)

When the four-force is zero one has gravitation acting alone, and the four-vector version of Newton's second law above reduces to the geodesic equation. A particle executing geodesic motion has zero value for each component of the acceleration four vector.This conforms to the fact that Gravity is not a force. The radial component four-acceleration of a body under free-fall,incidentally, is zero.This identifies the fact that the spatial part of four-acceleration is different from what we understand by acceleration in common day to day experience for example in the case of an apple falling from a tree.

In non-inertial coordinates, which include accelerated coordinates in special relativity and all coordinates in general relativity, the acceleration four-vector is related to the four-velocity through an absolute derivative with respect to proper time.

In inertial coordinates the Christoffel symbols are all zero, so this formula is compatible with the formula given earlier.

In special relativity the coordinates are those of a rectilinear inertial frame, so the Christoffel symbols term vanishes, but sometimes when authors use curved coordinates in order to describe an accelerated frame, the frame of reference isn't inertial, they will still describe the physics as special relativistic because the metric is just a frame transformation of the Minkowski space metric. In that case this is the expression that must be used because the Christoffel symbols are no longer all zero.